Multivariate polynomial inequalities of different \(L_{p,W}(V)\)-metrics with \(k\)-concave weights
type: Item
instance of: scholarly article
title: Multivariate polynomial inequalities of different \(L_{p,W}(V)\)-metrics with \(k\)-concave weights
zbMATH Open document ID: 1389.41014
published in: Acta Mathematica Hungarica
publication date: 2017-11-10T00:00:00Z
review text: Let \(W\) be a \(k\)-concave weight on an open convex set \(V\) in \(\mathbb {R} ^m,\;k\in [0,\infty]\), and let \(\mu_W\) be the weighted measure on \(V\) generated by \(W\) with \(\mu_W(V)<\infty \). Next, let \(L_{r,W}(V),\;0\leq r\leq \infty \), be the space of all Lebesque-masurable functions \(f:V\rightarrow \mathbb {R}^1\) on \(V\) with the finite functional \(\| f\|_{L_{r,W}(V)}:=(\mu_W (V)^{-1}\int_V| f(x)| ^rW(x)dx)^{1/r}\) for \(0<r<\infty \). For the case \(r=0\) and \(r=\infty \), this functional is naturally extended. By \(\mathcal{P}_{n,m}\) is denoted the set of all algebraic polynomials of \(m\) variables with real coefficients of degree at most \(n\). Lower and upper estimates of a constant \(A_k\) defined by \[ A_k(n,m, p,q):= \sup_V\sup_{W\in C_k(V)}\sup_{P\in \mathcal {P}_{n,m}} \| P\|_{L_{q,W}(V)}/\| P\|_{L_{p,W}(V)},\;0\leq p<q\leq \infty, \] are founded, where \(C_k(V)\) is the set of all \(k\)-concave weights on \(V\) with \(k\in [0,\infty]\). For \(k=0\in (0,\infty ]\), estimates of \(A_k\) are new and they are sharp with respect to \(n\) as \(n\rightarrow \infty \). For \(k=0\), the obtained by A. Brudnyi estimate is improved. Moreover, several examples of computation or estimation for \(\| P\|_{L_{q,W}(V)}/\| P\|_{L_{p,W}(V)}\) are presented.
reviewed by: D. K. Ugulava
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 26D05
zbMATH DE Number: 6806077
zbMATH Keywords: multivariate polynomial
zbMATH Keywords: polynomial inequality of different metrics
zbMATH Keywords: \((1/d)\)-concave measure
zbMATH Keywords: log-concave measure
author: Michael I. Ganzburg
MaRDI profile type: Publication
full work available at URL: s10474-016-0632-z
OpenAlex ID: W2462938409
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DOI: 10.1007/S10474-016-0632-Z
description: scientific article; zbMATH DE number 6806077